Compact Kähler manifolds with nonpositive bisectional curvature

Gang Liu

doi:10.1007/s00039-014-0290-7

Compact Kähler manifolds with nonpositive bisectional curvature

Gang Liu^*

^*Corresponding author for this work

University of California at Berkeley

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Let (Mⁿ, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N^k with c₁ < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

Original language	English
Pages (from-to)	1591-1607
Number of pages	17
Journal	Geometric and Functional Analysis
Volume	24
Issue number	5
DOIs	https://doi.org/10.1007/s00039-014-0290-7
State	Published - 1 Sep 2014
Externally published	Yes

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abstract = "Let (Mn, g) be a compact K{\"a}hler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K{\"a}hler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact K{\"a}hler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.",

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AB - Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

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JF - Geometric and Functional Analysis

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Compact Kähler manifolds with nonpositive bisectional curvature

Abstract

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